Does Throwing a Clock in Free-Fall Affect Time Dilation Calculations?

[cragar]

If I am in free-fall and then I throw a clock above me so that it is moving away from me
at a constant speed, can I just use regular time dilation to see how time is flowing in the clocks frame relative to my frame? Or do I need to worry about gravitational time dilation.

 

[HallsofIvy]

You and the clock, even after you have thrown it away from you (ignoring the momentary acceleration) are still in the same frame of reference and you will not see any change in the rate of the clock. An observer, at rest with respect to the planet, will see both the clock and you slowing down as you go deeper into the gravity well.

 

[cragar]

ok so being in free-fall is the same as floating in free space. So when I throw the clock it moves away. Why is this considered the same reference frame. How could we release this clock so it appears to move away from me at a constant speed and be in a different frame of reference with me still in free fall ?

 

[HallsofIvy]

I was referring to the vertical motion only. You are correct that the clock has non zero horizontal speed with respect to you- that has nothing to do with the gravity.

 

[yuiop]

If I am in free-fall and then I throw a clock above me so that it is moving away from me at a constant speed, can I just use regular time dilation to see how time is flowing in the clocks frame relative to my frame? Or do I need to worry about gravitational time dilation.

If you throw the clock upwards with velocity v, then it will time dilate by √(1-v^2/c^2). If the distance between you and the clock is Δr and the distance from the centre of gravitational mass is r, then if Δr/r is insignificant you can ignore gravitational time dilation. In other words, while you are in free fall the spacetime around you is aproximately flat Minkowskian in a small local region.

 

[A.T.]

You and the clock, even after you have thrown it away from you (ignoring the momentary acceleration) are still in the same frame of reference and you will not see any change in the rate of the clock.

I was referring to the vertical motion only.

You are still wrong. When you are in free fall and throw the clock vertically upwards it will continue to move away from you. You and the clock stay in relative motion, so of course there is time dilation from motion.

 

[yuiop]

If I am in free-fall and then I throw a clock above me so that it is moving away from me at a constant speed, can I just use regular time dilation to see how time is flowing in the clocks frame relative to my frame?

Short answer, yes. (In a local region.)

 

[pervect]

If I am in free-fall and then I throw a clock above me so that it is moving away from me
at a constant speed, can I just use regular time dilation to see how time is flowing in the clocks frame relative to my frame? Or do I need to worry about gravitational time dilation.

If you are in free fall near a massive body, such as the Earth, to the first order in (v/c), it will be as if you were in an inertial frame in which there was no gravitational time dilation.

You will see effects of second order in (v/c), which can be ascribed to tidal forces.

I have a feeling the detailed reasoning behind this would just confuse the thread, so I'll omit it unless asked. I will however state the results in a different way:

Gravitational time dilation can be thought of as the value of the metric coefficient g_00. In a free-fall frame, g_00 will be 1 and the first derivative of g_00 will be 0, but the second derivative of g_00 will be nonzero and proportional to the tidal forces in the free-falll frame.